ON THE METHOD OF MAXIMA AND MINIMA

...property The sentence begins mid-thought from the previous page: "...any other property [endowed with a maximum or minimum] is sought..." is sought among all curves that have one or more proposed properties in common; and from the number of these properties, a further subdivision of this treatise will arise. In the meantime, however, there will be no need to go further into this subdivision; since a method will soon be found for easily resolving problems, no matter how many properties have been proposed. For the solutions of problems that at first glance appear most intricate will, beyond expectation, become quite easy to handle and can be completed with a light calculation.

HYPOTHESIS I.

16. In this treatise, we shall perpetually designate the abscissa abscissa: the horizontal coordinate (x) on a graph, to which we relate all curves, by the letter x, *and the ordinate ordinate: the vertical coordinate (y) on a graph; the original Latin term is "applicata" by the letter y. Then indeed, assuming equal elements of the abscissa This means treating the change in x (dx) as a constant value., it will always be: dy = pdx; dp = qdx; dq = rdx; dr = sdx; etc.

COROLL. I.

17. By these substitutions, therefore, all differentials differentials: infinitely small changes in a variable, used here to build up higher-order derivatives like acceleration or curvature of y of whatever degree will be removed from the expressions, and no other differentials will be left besides the differential dx. Although in this way all differentials, except dx, are removed only in appearance and not in reality; nevertheless, these substitutions will provide us with immense assistance in the present undertaking.

COROLL. II.

18. Furthermore, by substitutions of this kind, the assumption of a constant differential is entirely removed from the calculation: for whatever other differential is assumed to be constant, the same formula must always emerge after these substitutions. In the meantime, however, because of the method to be employed below, it will be necessary to assume the differential dx as a constant.